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Soft and rigid impact
Amabile Tatone
Dipartimento di Ingegneria delle Strutture, delle Acque e del TerrenoUniversita dell’Aquila - Italy
1st International Conference on Computational Contact MechanicsLecce, Italy, Sept. 16-18, 2009
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Based on a joint work with:
Alessandro Contento and Angelo Di Egidio
Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno
Universita dell’Aquila - Italy
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References
I Alessandro Granaldi, Paolo Decuzzi, The dynamic response ofresistive microswitches: switching time and bouncing, J.Micromech. Microeng., 16, 2006.
I Z. J. Guo, N. E. McGruer, G. G. Adams, Modeling, simulation andmeasurement of the dynamic performance of an ohmic contact,electrostatically actuated RF MEMS switch, J. Micromech.Microeng., 17, 2007
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A toy model for contact simulations
Contact between a body and a rigid flat support
I rigid body
I affine body (homogeneous deformations)
I contractile affine body
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Contact force constitutive laws
Repulsive force:
qr (x, t) = αr d(x, t)−νr n
Damping force:
qd(x, t) = −βd d(x, t)−νd (n⊗ n) p(x, t)
Frictional force:
qf (x, t) = −βf d(x, t)−νf (I− n⊗ n) p(x, t)
Adhesive force:
qa(x, t) = −βa (d(x, t)−νaa − d(x, t)−νar ) n
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Contact force constitutive laws
repulsive forcerepulsive + adhesive forcesrepulsive + adhesive forces
d
q
νr = 8, νaa = 3, νar = 6
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Contact force constitutive laws
n
o
d(x, t)
d(x, t) := (p(x, t)− o) · n
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Contact force constitutive laws
n
o
d0
d(x, t) := (p(x, t)− o) · n
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Rigid block
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Rigid disk
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Numerical simulations (rigid body)
L
R
dL
dR
θ
t t
001 011
002 012
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Numerical simulations (rigid body)
rocking on a sloping plane
021 022 023
bouncing
031
rolling
032 033 034 035
adhesion and detachment501 502 503 505
spinning top
3D-101 3D-111
3D-102 3D-112
dice throwing
3D-201 3D-211
3D-202 3D-212
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Affine body
F = ∇p
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Affine body
The motion of a body B is described at each time t by a transplacementp(·, t) defined on the reference shape D :
p : D× I→ E
characterized by the following representation:
p(x, t) = p0(t) +∇p(t)(x− x0)
where ∇p(t) : V→ V is a tensor such that det∇p(t) > 0.An affine velocity field v at time t has the representation:
v(x) = v0 +∇v(x− x0)
Along a motion at time t
v0 = p0(t), ∇v = ∇p(t)
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Affine body
Balance principle:
∫D
b(x, t) · v dV +
∫∂D
q(x, t) · v dA− S(t) · ∇v vol(D) = 0 , ∀v
Balance equations:
−m p0(t)−m g + f(t) = 0
−∇p(t) J + M(t)− S(t) vol(D) = 0
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Mass and Euler tensor:
m :=
∫Dρ dV
J :=
∫Dρ(x− x0)⊗ (x− x0) dV
Total force and moment tensor:
f(t) :=
∫∂D
q(x, t) dA
M(t) :=
∫∂D
(x− x0)⊗ q(x, t)dA
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Material constitutive characterization
Frame indifference:
S ·WF = 0 ∀W | sym W = 0 ⇒ skw SFT = 0
Dissipation inequality:
S · F− d
dtϕ(F) ≥ 0
Reduced dissipation inequality:
S+FT · FF−1 ≥ 0
S+ := S− S(F)
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Material constitutive characterization
Hyperelastic stress:
S(F) · F =dϕ(F)
dt
Mooney-Rivlin strain energy (incompressible material):
ϕ(F) := c1(ı1(C)− 3) + c2(ı2(C)− 3) .
ı1(C) := tr (C), ı2(C) :=1
2
(tr(C)2 − tr(C2)
).
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Material constitutive characterization
Reduced dissipation inequality:
S+FT · FF−1 ≥ 0
S+ := S− S(F)
The simplest way to satisfy a-priori the dissipation inequality:
S+FT = µ sym (FF−1) , µ ≥ 0
Stress response (dissipative + energetic + reactive):
S = µ sym (FF−1)(FT)−1 + S0(F)− π (FT)−1
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Contact forces
Surface forces per unit deformed area:
q(x, t) =∑
j
qj(x, t) k(x, t)
Area change factor:
k(x, t) := ‖∇p(t)−Tn∂D(x)‖ det∇p(t)
n∂D(x) outward unit normal vector
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Numerical simulations (elastic body)
elastic bouncing, rolling and oscillations
041 112
200 214 215 216 217
318 319
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Affine contractile body
∇p
G
F
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Affine contractile body
∇p
GF
Kroner-Lee decomposition:
F(t) := ∇p(t) G(t)−1
Contraction velocity:
V = GG−1
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Affine contractile body
Balance principle:
∫D
b(x, t) · v dV +
∫∂D
q(x, t) · v dA− S(t) · ∇v vol(D)
+(Q(t) · V − A(t) · V
)vol(D) = 0 , ∀(v,V)
Balance equations:
−m p0(t)−m g + f(t) = 0
−∇p(t) J + M(t)− S(t) vol(D) = 0
Q(t)− A(t) = 0
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Material constitutive characterization
Frame indifference:
S ·W∇p = 0 ∀W | sym W = 0 ⇒ skw S∇pT = 0
Dissipation inequality:
A · GG−1 + S · ∇p− d
dt
(ϕ(F) det G
)≥ 0
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Material constitutive characterization
Reduced dissipation inequality:
S+∇pT · FF−1 + A+ · GG−1 ≥ 0
S+ := S− S(F) , A+ := A + FTSGT − (det G)ϕ(F)I
Hyperelastic stress:
S(F)GT · F =dϕ(F)
dt
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Material constitutive characterization
The simplest way to satisfy a-priori the dissipation inequality:
S+∇pT = µ sym (FF−1) , µ ≥ 0
A+ = µγ GG−1 , µγ ≥ 0
Stress characterization:
S = µ sym (FF−1)(∇pT)−1 + S0(F)− π (∇pT)−1
A = µγ GG−1 −(FTSGT − (det G)ϕ(F)I
)
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Material constitutive characterization
Equations of motion:
−m p0 −m g + f = 0
−∇p J + M− S vol(D) = 0
µγ GG−1 = FTSGT − (det G)ϕ(F)I + Q
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Numerical simulations (contractile body)
oscillating driving Q
12g1 12g2 12g3
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Numerical simulations (contractile body)
oscillating driving G
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Numerical simulations (contractile body)
oscillating driving G
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References
I Simo J.C., Wriggers P. and Taylor R.L., “A perturbed Lagrangianformulation for the finite element solution of contact problems,”Comp. Methods Appl. Mech. Engrg., 51, 163–180 (1985).
I Wriggers-Za Wriggers P. and Zavarise G., “Chapter 6,Computational Contact Mechanics,” in Encyclopedia ofComputational Mechanics, Stein E., de Borst R., Hughes T.J.R.,editors, John Wiley & Sons (2004).
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References
I Di Carlo A. and Quiligotti S., “Growth and Balance,” Mech. Res.Comm., 29, 449–456 (2002).
I Di Carlo A., “Surface and bulk growth unified,” in Mechanics ofMaterial Forces, Steinmann P. and Maugin G. A., editors, Springer,New York, 53–64 (2005).
I Nardinocchi P. and Teresi L., “On the active response of soft livingtissues,” J. Elasticity, 88, 27–39 (2007).
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Supplementary references
I Gianfranco Capriz, Paolo Podio-Guidugli, Whence the boundaryconditions in modern continuum physics?, Atti Convegni Lincei n.210, 2004
I Antonio Di Carlo, Actual surfaces versus virtual cuts, Atti ConvegniLincei n. 210, 2004
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